Shifted log-logistic distribution

The shifted log-logistic distribution can be obtained from the log-logistic distribution by addition of a shift parameter ${\displaystyle \delta }$. Thus if ${\displaystyle X}$ has a log-logistic distribution then ${\displaystyle X+\delta }$ has a shifted log-logistic distribution. So ${\displaystyle Y}$ has a shifted log-logistic distribution if ${\displaystyle \log(Y-\delta )}$ has a logistic distribution. The shift parameter adds a location parameter to the scale and shape parameters of the (unshifted) log-logistic.

The properties of this distribution are straightforward to derive from those of the log-logistic distribution. However, an alternative parameterisation, similar to that used for the generalized Pareto distribution and the generalized extreme value distribution, gives more interpretable parameters and also aids their estimation.

In this parameterisation, the cumulative distribution function (CDF) of the shifted log-logistic distribution is

${\displaystyle F(x;\mu ,\sigma ,\xi )={\frac {1}{1+\left(1+{\frac {\xi (x-\mu )}{\sigma }}\right)^{-1/\xi }}}}$

for ${\displaystyle 1+\xi (x-\mu )/\sigma \geqslant 0}$, where ${\displaystyle \mu \in \mathbb {R} }$ is the location parameter, ${\displaystyle \sigma >0\,}$ the scale parameter and ${\displaystyle \xi \in \mathbb {R} }$ the shape parameter. Note that some references use ${\displaystyle \kappa =-\xi \,\!}$ to parameterise the shape.[3][4]

The probability density function (PDF) is

${\displaystyle f(x;\mu ,\sigma ,\xi )={\frac {\left(1+{\frac {\xi (x-\mu )}{\sigma }}\right)^{-(1/\xi +1)}}{\sigma \left[1+\left(1+{\frac {\xi (x-\mu )}{\sigma }}\right)^{-1/\xi }\right]^{2}}},}$

again, for ${\displaystyle 1+\xi (x-\mu )/\sigma \geqslant 0.}$

The shape parameter ${\displaystyle \xi }$ is often restricted to lie in [-1,1], when the probability density function is bounded. When ${\displaystyle |\xi |>1}$, it has an asymptote at ${\displaystyle x=\mu -\sigma /\xi }$. Reversing the sign of ${\displaystyle \xi }$ reflects the pdf and the cdf about ${\displaystyle x=\mu .}$.